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Question

Which of the following is/are true, if z, z1, z2 are complex numbers?


  1. All of these


Solution

The correct options are
A


B


Let z = r (cosθ+isinθ)

¯¯¯¯Z = r (cosθisinθ)

Option A: (¯¯¯¯¯¯¯¯Z) = r (cosθ+isinθ) = z 

Option B: znrn (cosθ+isinθ)n

zn = r (cosθ+isinθ) {Use de moivre's theorem (cosθ+isinθ)n = (cosθ+isinθ) }

(¯¯¯¯¯¯¯Zn) = rn (cosnθ+isinnθ)

(¯¯¯¯Z)n =  rn (cosθisinθ)n

(¯¯¯¯Z)n =  rn (cosnθ+isinnθ) { Use de moivre's theorem (cosθ+isinθ)n = (cosθisinθ) }

So, (¯¯¯¯¯¯¯Zn) = (¯¯¯¯Z)n

Let z1r1 (cosθ1+isinθ1) = r1 eiθ1

      z2r2 (cosθ2+isinθ2) = r2 eiθ2

      z1.z2 = r1.r2 [(cos(θ1+θ2)+isin(θ1+θ2))]

      (¯¯¯¯¯¯¯¯¯¯¯¯z1.z2) = r1.r2 [(cos(θ1+θ2)isin(θ1+θ2))]

      So, We see that

      (¯¯¯¯¯¯¯¯¯¯¯¯z1.z2) z1.z2

      Only option A and B are correct.

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